Sutton and Barto 2nd Edition Exercise 13.1
There are problems with both the approach and the expressions that you have. I don't want to just give the correct solution, though, that's an exercise for you to go through and learn from your own experience trying to accomplish it. Instead, let me illustrate that your expressions for $v(s_i)$ are wrong. To do that we'll just do a Monte-Carlo estimate for a range of values of $p$ and compare to your expression.
Here's the Python code that runs a single payout starting from state state and following the policy with probability p of choosing &quot;right&quot;. It returns the collected reward:
import numpy as np

def playout(state, p):
   reward = 0
   while state != 0:
       action_right = np.random.rand() &lt; p
       move_right = action_right if state != 2 else (not action_right)
       state = state - 1 if move_right else state + 1
       state = 3 if state &gt; 3 else state
       reward -= 1
   return reward

Then we make a simulation code that runs the playout multiple times, collects the reward counts, and returns the average reward (so, essentially estimates $v^\pi(s_i)$):
from collections import defaultdict

def simulate(state, p, n):
    rewards = defaultdict(lambda : 0, {})
    for _ in range(n):
        rewards[playout(state,p)] += 1
    results = np.array([[a,b] for a,b in rewards.items()]).T
    reward , nplayouts = results[0] , results[1]
    value = (reward * nplayouts).sum() / nplayouts.sum()
    return value

Finally, I make a grid in p and run each playout 10000 times:
p = np.linspace(0,1,51)[1:-1]
v3 = [simulate(3,p,10000) for p in p]
v2 = [simulate(2,p,10000) for p in p]
v1 = [simulate(1,p,10000) for p in p]

I've plotted the resulting value estimates for each state. Together with your expression for them (blue curves). And the correct expression (red curve) that I've obtained by actually writing down and solving the equations:

As you can see, the expressions you've presented are all too far off from the results that the simulation returns. More than that - the asymptotic behavior of your solutions at $p\to0$ and $p\to1$ doesn't make much sense.
I've obtained the expressions for the red curves above by solving the system for $v(s_i)$ on my own account - without relying on weird and wrong solutions that I googled on the internet. Which I'd recommend you do as well.
Finally, the question of the exercise is to find an optimal $p$ for a policy that starts at $s_3$ - not &quot;irrespective of starting state&quot; as you've thought is should be. Unlike your expression the correct expression for  $v(s_3)$ has a maximum, which can be found analytically and it is
$$\max_p v(s_3) = -6-4\sqrt2 \simeq -11.6$$
$$ \text{at}\quad p = ??? \simeq 0.59$$